Method for introducing dependence of white-box implementation on a set of strings

ABSTRACT

A method of performing a cryptographic operation using a cryptographic implementation in a cryptographic system, including: receiving, by the cryptographic system, an identifying string value; receiving, by the cryptographic system, an input message; performing, by the cryptographic system, a keyed cryptographic operation mapping the input message into an output message wherein the output message is the correct result when the identifying string value is one of a set of binding string values, wherein the set includes a plurality of binding string values.

RELATED APPLICATION

This patent application is related to U.S. patent application Ser. No.14/263,429, entitled “METHOD FOR INCLUDING AN IMPLICIT INTEGRITY ORAUTHENTICITY CHECK INTO A WHITE-BOX IMPLEMENTATION” to Michiels et al.(hereinafter “related application”).

TECHNICAL FIELD

Various exemplary embodiments disclosed herein relate generally to amethod for introducing dependence of a white-box implementation on a setof strings.

BACKGROUND

The Internet provides users with convenient and ubiquitous access todigital content. Because the Internet is a powerful distributionchannel, many user devices strive to directly access the Internet. Theuser devices may include a personal computer, laptop computer, set-topbox, internet enabled media player, mobile telephone, smart phone,tablet, mobile hotspot, or any other device that is capable of accessingthe Internet. The use of the Internet as a distribution medium forcopyrighted content creates the compelling challenge to secure theinterests of the content provider. Increasingly, user devices operateusing a processor loaded with suitable software to render (playback)digital content, such as audio and/or video. Control of the playbacksoftware is one way to enforce the interests of the content ownerincluding the terms and conditions under which the content may be used.Previously many user devices were closed systems. Today more and moreplatforms are partially open. Some users may be assumed to have completecontrol over and access to the hardware and software that providesaccess to the content and a large amount of time and resources to attackand bypass any content protection mechanisms. As a consequence, contentproviders must deliver content to legitimate users across a hostilenetwork to a community where not all users or user devices can betrusted.

Secure software applications may be called upon to carry out variousfunctions such as, for example, cryptographic functions used to protectand authenticate digital content. In order to counter attacks, thesealgorithms have to be obfuscated (hidden) in order to prevent reverseengineering and modification of the algorithm or prohibit obtaining theuser-specific secure information. Accordingly, the functions of thesecure software application may be carried out by various functions asdefined by the instruction set of the processor implementing the securesoftware. For example, one way to obscure these functions is by the useof lookup tables.

The widespread use of digital rights management (DRM) and other securesoftware has given rise to the need for secure, tamper-resistantsoftware that seeks to complicate tampering with the software. Varioustechniques for increasing the tamper resistance of software applicationsexist. Most of these techniques are based on hiding the embeddedknowledge of the application by adding a veil of randomness andcomplexity in both the control and the data path of the softwareapplication. The idea behind this is that it becomes more difficult toextract information merely by code inspection. It is therefore moredifficult to find the code that, for example, handles access andpermission control of the secure application, and consequently to changeit.

As used herein, white-box cryptography includes a secure softwareapplication that performs cryptographic functions in an environmentwhere an attacker has complete control of the system running thewhite-box cryptography software. Thus, the attacker can modify inputsand outputs, track the operations of the software, sample and monitormemory used by the software at any time, and even modify the software.Accordingly, the secure functions need to be carried out in a mannerthat prevents the disclosure of secret information used in the securefunctionality. White-box cryptography functions may be implemented invarious ways. Such methods include: obscuring the software code; usingcomplex mathematical functions that obscure the use of the secretinformation; using look-up tables; using finite state machines; or anyother methods that carry out cryptographic functions but hide the secretinformation needed for those secure functions. A white-boximplementation may also contain components that include anti-debuggingand tamper-proofing properties.

There are several reasons for preferring a software implementation of acryptographic algorithm to a hardware implementation. This may, forinstance, be the case because a software solution is renewable if thekeys leak out, because it is has lower cost, or because theapplication-developer has no influence on the hardware where thewhite-box system is implemented.

SUMMARY

A brief summary of various exemplary embodiments is presented below.Some simplifications and omissions may be made in the following summary,which is intended to highlight and introduce some aspects of the variousexemplary embodiments, but not to limit the scope of the invention.Detailed descriptions of an exemplary embodiment adequate to allow thoseof ordinary skill in the art to make and use the inventive concepts willfollow in later sections.

Various exemplary embodiments relate to a non-transitorymachine-readable storage medium encoded with instructions for executionby a cryptographic implementation in a cryptographic system forperforming a cryptographic operation, the non-transitorymachine-readable storage medium including: instructions for receiving,by the cryptographic system, an identifying string value; instructionsfor receiving, by the cryptographic system, an input message;instructions for performing, by the cryptographic system, a keyedcryptographic operation mapping the input message into an output messagewherein the output message is the correct result when the identifyingstring value is one of a set of binding string values, wherein the setincludes a plurality of binding string values.

Various embodiments are described wherein there are input messages forwhich the output message is an incorrect result when the identifyingstring is not in the set of binding string values.

Various embodiments are described wherein the identifying string valueis based upon an identification of the cryptographic implementation.

Various embodiments are described wherein the identifying string valueis based upon a hash of a portion of code in the cryptographic system.

Various embodiments are described wherein the identifying string valueis based upon an identification of the cryptographic system.

Various embodiments are described wherein the identifying string valueis based upon a user password.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the non-transitory machine-readablestorage medium further includes: instructions for encoding an output ofthe first function based upon the identifying string value; andinstructions for performing the second function on the encoded output ofthe first function wherein the second function includes decoding theencoded output of the first function using the set of binding stringvalues.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the non-transitory machine-readablestorage medium further includes: instructions for encoding an output ofthe first function using the set of binding string values; andinstructions for performing the second function on the encoded output ofthe first function wherein the second function includes decoding theencoded output of the first function based upon the identifying stringvalue.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the non-transitory machine-readablestorage medium further includes: instructions for perturbing an outputof the first function using the identifying string value; andinstructions for performing the second function on the perturbed outputof the first function wherein the second function includes compensatingfor the perturbation of the output of the first function using the setof binding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the non-transitory machine-readablestorage medium further includes: instructions for introducing aperturbation in the calculation of the first function based upon theidentifying string value; instructions for compensating for theperturbation in the calculation of the first function during calculationof the second function based upon the set of binding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the non-transitory machine-readablestorage medium further includes: instructions for introducing aperturbation in the calculation of the first function based upon the setof binding string values; instructions for compensating for theperturbation in the calculation of the first function during calculationof the second function based upon the identifying string value.

Various embodiments are described wherein the cryptographic systemincludes a network of finite state machines.

Various embodiments are described wherein the cryptographic systemincludes a network of lookup tables.

Various embodiments are described wherein the cryptographic operation isone of advanced encryption system (AES) or data encryption standard(DES).

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the non-transitory machine-readablestorage medium further includes: instructions for modifying the firstfunction based upon the identifying string value.

Various embodiments are described wherein the first function is aplurality of instructions, and modifying the first function based uponthe identifying string value includes producing a modifying string valuebased upon the identifying string value and wherein a portion of theplurality of instructions is implemented using the modifying stringvalue.

Further, various exemplary embodiments relate to a method of producing acryptographic implementation of a cryptographic operation mapping aninput message to an output message in a cryptographic system that bindsthe cryptographic implementation to a binding string value, including:receiving information specifying a set of binding string values;modifying a cryptographic implementation to receive an identifyingstring value; modifying the cryptographic implementation based upon thereceived information specifying the set of binding string values sothat: when a received identifying string value is one of the set ofbinding string values, the cryptographic implementation outputs acorrect output message.

Various embodiments are described wherein when a received identifyingstring value is not one of the set of binding string values, thecryptographic implementation outputs an incorrect output message.

Various embodiments are described wherein the identifying string valueis based upon an identification of the cryptographic implementation.

Various embodiments are described wherein the identifying string valueis based upon a hash of a portion of the cryptographic implementation.

Various embodiments are described wherein the identifying string valueis based upon an identification of the cryptographic system.

Various embodiments are described wherein the identifying string valueis based upon a user password.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and modifying the cryptographicimplementation so that: the output of one of the first function isencoded based upon the identifying string value; and a second functionis performed on the encoded output of the first function wherein thesecond function includes decoding the encoded output of the firstfunction using the set of binding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and modifying the cryptographicimplementation so that: the output of one of the first function isencoded based upon the set of binding string values; and a secondfunction is performed on the encoded output of the first functionwherein the second function includes decoding the encoded output of thefirst function based upon the identifying string value.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and modifying the cryptographicimplementation so that: the output of one of the first function isperturbed using the identifying string value; and a second function isperformed on the encoded output of the first function wherein the secondfunction includes compensating for the perturbation of the output of thefirst function using the set of binding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and modifying the cryptographicimplementation so that: a perturbation is introduced in the calculationof the first function based upon the identifying string value; and theperturbation in the calculation of the first function is compensated forduring calculation of the second function based upon the set of bindingstring values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and modifying the cryptographicimplementation so that: a perturbation is introduced in the calculationof the first function based upon the set of binding string values; theperturbation in the calculation of the first function is compensated forduring calculation of the second function based upon the identifyingstring value.

Various embodiments are described wherein the cryptographic systemincludes a network of finite state machines.

Various embodiments are described wherein the cryptographic systemincludes a network of lookup tables.

Various embodiments are described wherein the cryptographic operation isone of advanced encryption system (AES) or data encryption standard(DES).

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction, and modifying the cryptographic implementation so that: thefirst function is based upon the identifying string value.

Various embodiments are described wherein the first function is aplurality of instructions, and modifying the cryptographicimplementation so that the first function is based upon the identifyingstring value includes producing a modifying string value based upon theidentifying string value and wherein a portion of the plurality ofinstructions is implemented using the modifying string value.

Further, various exemplary embodiments relate to a method of performinga cryptographic operation using a cryptographic implementation in acryptographic system, including: receiving, by the cryptographic system,an identifying string value; receiving, by the cryptographic system, aninput message; performing, by the cryptographic system, a keyedcryptographic operation mapping the input message into an output messagewherein the output message is the correct result when the identifyingstring value is one of a set of binding string values, wherein the setincludes a plurality of binding string values.

Various embodiments are described wherein there are input messages forwhich the output message is an incorrect result when the identifyingstring is not in the set of binding string values.

Various embodiments are described wherein the identifying string valueis based upon an identification of the cryptographic implementation.

Various embodiments are described wherein the identifying string valueis based upon a hash of a portion of code in the cryptographic system.

Various embodiments are described wherein the identifying string valueis based upon an identification of the cryptographic system.

Various embodiments are described wherein the identifying string valueis based upon a user password.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the method further includes: encodingan output of the first function based upon the identifying string value;and performing the second function on the encoded output of the firstfunction wherein the second function includes decoding the encodedoutput of the first function using the set of binding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the method further includes: encodingan output of the first function using the set of binding string values;and performing the second function on the encoded output of the firstfunction wherein the second function includes decoding the encodedoutput of the first function based upon the identifying string value.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and a second function, and the method further includes:perturbing an output of the first function using the identifying stringvalue; and performing the second function on the perturbed output of thefirst function wherein the second function includes compensating for theperturbation of the output of the first function using the set ofbinding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the method further includes:introducing a perturbation in the calculation of the first functionbased upon the identifying string value; compensating for theperturbation in the calculation of the first function during calculationof the second function based upon the set of binding string values.

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction and second function, and the method further includes:introducing a perturbation in the calculation of the first functionbased upon the set of binding string values; compensating for theperturbation in the calculation of the first function during calculationof the second function based upon the identifying string value.

Various embodiments are described wherein the cryptographic systemincludes a network of finite state machines.

Various embodiments are described wherein the cryptographic systemincludes a network of lookup tables.

Various embodiments are described wherein the cryptographic operation isone of advanced encryption system (AES) or data encryption standard(DES).

Various embodiments are described wherein the cryptographicimplementation includes a plurality of functions including a firstfunction, and the method further includes: modifying the first functionbased upon the identifying string value.

Various embodiments are described wherein the first function is aplurality of instructions, and modifying the first function based uponthe identifying string value includes producing a modifying string valuebased upon the identifying string value and wherein a portion of theplurality of instructions is implemented using the modifying stringvalue.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to better understand various exemplary embodiments, referenceis made to the accompanying drawings, wherein:

FIG. 1 illustrates the main steps of a round of AES;

FIG. 2 illustrates a white-box AES implementation with encodings on theinput of the rounds;

FIG. 3 illustrates the computation of one output nibble by means of anetwork of look-up tables;

FIG. 4 illustrates a portion of the network table of FIG. 3 obfuscatedby encoding the inputs and outputs; and

FIG. 5 illustrates a first embodiment of binding a white-boximplementation;

FIG. 6 illustrates the application of obfuscation to the white-boximplementation of FIG. 5;

FIG. 7 illustrates a second embodiment of binding a white-boximplementation; and

FIG. 8 is a flow chart illustrating a method of binding a white-boximplementation to a set of binding strings.

To facilitate understanding, identical reference numerals have been usedto designate elements having substantially the same or similar structureand/or substantially the same or similar function.

DETAILED DESCRIPTION

The description and drawings illustrate the principles of the invention.It will thus be appreciated that those skilled in the art will be ableto devise various arrangements that, although not explicitly describedor shown herein, embody the principles of the invention and are includedwithin its scope. Furthermore, all examples recited herein areprincipally intended expressly to be for pedagogical purposes to aid thereader in understanding the principles of the invention and the conceptscontributed by the inventor(s) to furthering the art, and are to beconstrued as being without limitation to such specifically recitedexamples and conditions. Additionally, the term, “or,” as used herein,refers to a non-exclusive or (i.e., and/or), unless otherwise indicated(e.g., “or else” or “or in the alternative”). Also, the variousembodiments described herein are not necessarily mutually exclusive, assome embodiments can be combined with one or more other embodiments toform new embodiments.

Code lifting is a problem that may arise with a software implementationof cryptographic algorithm. This problem may be overcome by binding thewhite-box implementation to an arbitrary given string s, i.e., thewhite-box implementation only works properly if the string s isavailable. Binding the white-box implementation to a string s, may beused to implement node locking, software tamper resistance, userbinding, and traitor tracing. The problem of code lifting may arisebecause a software implementation of a cryptographic algorithm may becopied and used on unauthorized nodes or by unauthorized users.Accordingly, a white-box implementation, although it may effectivelyhide a cryptographic key, may still be distributed as a whole. Thiswhite-box implementation may be as valuable as the key itself. If, forinstance, the white-box implementation implements a decryptionalgorithm, then by not having the key, the receiver may not be able toencrypt, but the white-box implementation is sufficient to decrypt. Thismeans that an adversary illegitimately distributes the white-boximplementation as a whole instead of the underlying hidden cryptographickey, which typically is of high value that should not be distributed inan uncontrolled way. Typically, the key is only present implicitly. Inother embodiments, the key may include dynamic keys that, for example,take implicit key information and change it with some sort of dynamicinformation to change the key used in the cryptographic function.

As mentioned above the use of an arbitrary given string s may be used toovercome these problems. A solution for using an arbitrary given strings is described in U.S. Pat. No. 8,479,016 to Michiels (“Michiels”). Thestring s may be some binary string that may be derived from the deviceon which the white-box implementation should be running and that cannotbe derived on other devices. For instance, s may be defined as a uniqueidentifier of the device. In Michiels a white-box implementation isderived containing s, and this white-box implementation is placed on thedevice where the string s is omitted where the string is part of alookup table or operational code used in the white-box implementation.As a result, the device can only execute the white-box implementationonce it can derive the omitted string s and then uses the value of swhen the table entry is requested that was omitted or to provide thecorrect instruction in the operational code, i.e., the white-boximplementation can only be executed on the legitimate device.

In the related application, embodiments are described that implement analternative approach for binding a white-box implementation to anarbitrary string s other than including it in the definition of a lookuptable. Instead the string s is a parameter of a function of thewhite-box implementation, i.e., the string s is not used for thespecification of the function (e.g., a lookup table or operational code)as in Michiels, but rather as a parameter of the function.

In the related application, a method for binding a white-boximplementation to an arbitrary string s is described. This descriptionis included below. Then the embodiments of Michiels and the relatedapplication will be extended to allow binding of the white-boximplementation to a plurality of strings.

Any strings s that represent the integrity or authenticity of the systemthat the white-box implementation runs on may be used. If, for instance,s is the hash of some code fragment, then it hardens the softwareagainst tampering. Also, the string s may be used for node locking wheres is the unique identification of a node or hardware in the white-boxsystem. Another use-case for the string s is traitor tracing. If thestring is the name of the legitimate owner of a white-boximplementation, and if this white-box implementation is encountered onan illegitimate device or on the Internet, then the white-boximplementation may be traced back to the source of the leakage. Further,s may be used to identify users by being associated with a user passwordor hash of a user password.

Both the related application and Michiels describe a method of makingthe correct functioning of a white-box implementation dependent on anarbitrary given string s. This may be used to solve the above problem ofcode lifting in the following way. Let the string s be some binarystring that may be derived from the device on which the white-boximplementation should be running and that cannot be derived on otherdevices. For instance, s may be defined as a unique identifier of thedevice. Next, the approach of the related application or Michiels isused to make the white-box implementation dependent on s. Then theobtained white-box implementation may be installed on a device, wherestring s is omitted. Then, the device may only execute the white-boximplementation once the device can derive the omitted string s, i.e.,the white-box implementation may only be executed on the legitimatedevice. Analogously, the white-box implementation may be bound to aperson by letting the string s be a password or a (digitized)fingerprint.

A property of both approaches is that once the white-box implementationis derived, there is no longer a choice in the string s that needs to beprovided to the implementation in order for it to work properly: thatis, it works properly for all messages if and only if string s ispresent. This is, for instance, sufficient if a white-box implementationis to be bound to a single device. Suppose, however, that a user is tobe provided with a white-box implementation that they can run onmultiple devices: for instance, their mobile phone and their tablet.Then for both approaches the user must be provided with two white-boximplementations: one for the mobile phone and one for the tablet.Another example where the option to choose between multiple strings inorder to get a properly functioning white-box implementation is thefollowing. Suppose that a white-box implementation is bound to a uservia a string s representing a password. Then forgetting the passwordmakes the white-box implementation useless. When the white-boximplementation is used to decrypt valuable data, this then means thatthe valuable data cannot be read anymore. A solution may include(besides the password string s) that the white-box implementation alsoworks properly for a password that is, for instance, only known by ahelpdesk. In that case, the helpdesk may help to restore the valuabledata. The embodiments of the invention described below make a white-boximplementation dependent on a set of strings in the sense that theimplementation works properly for any string in the set (instead of allstrings in the set).

In order to demonstrate embodiments of the invention, an examplewhite-box implementation of AES will now be described. White-boxcryptography is the discipline of implementing a cryptographic algorithmin software such that it is difficult for an attacker to find the key.Hereby, the strongest conceivable (but for software most realistic)attack model is assumed in which the adversary is assumed to have fullcontrol over and full access to the white-box implementation.

A table-based approach to a white-box implementation of the AdvancedEncryption Standard (AES) and the Data Encryption Standard (DES) wereproposed in the following papers: “White-Box Cryptography and an AESImplementation”, by Stanley Chow, Philip Eisen, Harold Johnson, and PaulC. Van Oorschot, in Selected Areas in Cryptography: 9th AnnualInternational Workshop, SAC 2002, St. John's, Newfoundland, Canada, Aug.15-16, 2002, referred to hereinafter as “Chow 1”; and “A White-Box DESImplementation for DRM Applications”, by Stanley Chow, Phil Eisen,Harold Johnson, and Paul C. van Oorschot, in Digital Rights Management:ACM CCS-9 Workshop, D R M 2002, Washington, D.C., USA, Nov. 18, 2002,referred to hereinafter as “Chow 2”. Chow 1 and Chow 2 disclose methodsof using a table-based approach to hide the cryptographic key by acombination of encoding its tables with random bijections, and extendingthe cryptographic boundary by pushing it out further into the containingapplication.

As noted, for many cryptographic operations it is desired to have awhite-box implementation. The invention may be applied, for example, tosymmetric and asymmetric cryptographic operations. Also, the inventionmay be applied to block ciphers, stream ciphers, message authenticationschemes, signature schemes, etc. Note that the invention may also beapplied to hash functions. The latter is especially useful if the hashfunction is used as a building block which processes secret information,e.g., a secret key, secret data, etc. For example, the invention may beapplied to a hash function used in a keyed-Hash Message AuthenticationCode (HMAC or KHMAC). Well known block ciphers include: AdvancedEncryption Standard (AES), Secure And Fast Encryption Routine, (SAFER,and variants SAFER+ and SAFER++), Blowfish, Data Encryption Standard(DES), etc. A well known stream cipher is RC4. Moreover any block ciphercan be used as stream cipher using an appropriate mode of operation,e.g., Cipher feedback (CFB), Counter mode (CTR), etc.

The white-box implementation may be implemented using a plurality ofbasic blocks. The plurality of basic blocks is interconnected, in thesense that some of the blocks build on the outputs of one or more of theprevious blocks. A basic block may also be implemented in softwarerunning on a general purpose computer chip, e.g. a microprocessor. Forexample, a basic block may use a plurality of computer instructions,including arithmetical instructions, which together implement thefunctionality of the basic block. A widely used implementation for thebasic block is a look-up table. For example, Chow 1 and Chow 2 take thisapproach to implement the AES and DES block ciphers. A look-up tableimplementation includes a list which lists for possible input values, anoutput value. The input value may be explicit in the lookup table. Inthat situation the look-up table implementation could map a particularinput to a particular output by searching in the list of input valuesfor the particular input. When the particular input is found theparticular output is then also found. For example, the particular outputmay be stored alongside the particular input. Preferably, the inputvalues are not stored explicitly, but only implicitly. For example, ifthe possible inputs are a consecutive range, e.g. of numbers orbit-strings, the look-up table may be restricted to storing a list ofthe output values. A particular input number may, e.g., be mapped to theparticular output which is stored at a location indicated by the number.Further, finite state machines or code obfuscation may be used toimplement the white-box implementation.

For example, a look up table for a function may be created by computingthe output value of the function for its possible inputs and storing theoutputs in a list. If the function depends on multiple inputs theoutputs may be computed and stored for all possible combinations of themultiple inputs. Look-up tables are especially suited to implementnon-linear functions, which map inputs to output in irregular ways. Awhite-box implementation can be further obfuscated, as is explainedbelow, by applying to one or more of its look-up tables a fixedobfuscating input encoding and a fixed output encodings. The results ofapplying a fixed obfuscating input encoding and output encodings is thenfully pre-evaluated. Using this technique, a look-up table would bereplaced by an obfuscated look-up table which has the same dimensions,that it takes the same number input bits and produces the same number ofoutput bits. The input encoding and output encoding used in suchobfuscation are not explicit in the final white-box implementation.

The network of basic blocks are arranged to compute an output messagewhen they are presented with an input message. Typically, the inputmessage is operated upon by a number of basic input blocks. A number offurther basic blocks may take input from one or more of the basic inputblocks and/or from the input. Yet further basic blocks can take input inany combination of the input message, the output of basic input blocksand the output of the further basic blocks. Finally some set of basicexit blocks, i.e., at least one, produce as output all or part of theoutput-message. In this manner a network of basic blocks emerges whichcollectively computes the mapping from the input message to outputmessage.

The key used may be a cryptographic key and may contain sufficiententropy to withstand an anticipated brute force attack. It is noted thatin a white-box implementation, the key is typically not explicitlypresent in the implementation. This would risk the key being found byinspection of the implementation. Typically, the key is only presentimplicitly. Various ways are known to hide a key in a cryptographicsystem. Typically, at least the method of partial evaluation is used,wherein a basic block which needs key input is evaluated in-so-far thatit does not depend on the input-message. For example, a basic operationwherein an input-value, a masking value, which does not depend on theinput-message, e.g. a value from an S-box, and a key-value need to beXORed can be partially evaluated by XORing the key value and the maskingvalue together beforehand. In this way the operation still depends onthe key-value although the key-value is not explicitly present in theimplementation. Instead, only the XOR between the key-value andmasking-value is present in the implementation. Note that, morecomplicated ways and/or further ways of hiding the keys are compatiblewith this invention.

Below exemplary embodiments are described using the AES (AdvancedEncryption Standard) block cipher, because AES has become a widely usedstandard for block ciphers. AES is a block cipher with a block size of128 bits or 16 bytes. The plaintext is divided in blocks of 16 byteswhich form the initial state of the encryption algorithm, and the finalstate of the encryption algorithm is the cipher text. At any given pointin the encryption algorithm these 16 bytes are the state of theencryption algorithm. To conceptually explain AES, the bytes of thestate are organized as a matrix of 4×4 bytes. AES includes a number ofrounds, which depends on the key size. Each round is includes similarprocessing steps operating on bytes, rows, or columns of the statematrix, each round using a different round key in these processingsteps. In the discussion using AES as an example, it is noted that AESdefines a round in a specific manner. In the embodiments below, a roundis any grouping of steps that includes at least one non-linear mappingfunction, such as an S-box in AES. Accordingly, a round as describedbelow includes one non-linear mapping function and any combination ofother steps of the cryptographic function.

FIG. 1 illustrates some main processing steps of a round of AES. Theprocessing steps include:

AddRoundKey 110—each byte of the state is XORed with a byte of the roundkey;

SubBytes 120—a byte-to-byte permutation using a lookup table;

ShiftRows 140—each row of the state is rotated a fixed number of bytes;and

MixColumns 150—each column is processed using a modulo multiplication inGF(2⁸).

The steps SubBytes 120, ShiftRows 130, and MixColumns 150 areindependent of the particular key used. The key is applied in the stepAddRoundKey 110. Except for the step ShiftRows 140, the processing stepscan be performed on each column of the 4×4 state matrix withoutknowledge of the other columns. Therefore, they can be regarded as32-bit operations as each column consists of four 8-bit values. Dashedline 150 indicates that the process is repeated until the requirednumber of rounds has been performed.

Each of these steps or a combination of steps may be represented by alookup table or by a network of lookup tables. If the AddRoundKey 110step is implemented by XORing with the round key, then the key isvisible to the attacker in the white-box attack context. The AddRoundKey110 step can also be embedded in lookup tables, which makes it lessobvious to find out the key. In fact, it is possible to replace a fullround of AES by a network of lookup tables. For example, the SubBytes120, ShiftRows 130, and MixColumns 150 steps may be implemented usingtable lookups. Below a possible white-box implementation of AES insufficient detail is discussed to describe the embodiments of theinvention below, but further detailed descriptions of such animplementation are found in Chow 1. Also, other variations in the lookuptable implementation may be used which are within the scope of theinvention.

Both the table-based white-box implementations and the finite statemachine implementations have the property that all intermediate valuesin the implementation are encoded (as compared to a standardimplementation). Examples of white-box implementations using finitestate machines are disclosed in U.S. Patent Publication 2007/0014394entitled “Data Processing Method” and a presentation at the Re-trustSixth Quarterly Meeting entitled “Synchrosoft MCFACT™ Secure DataProcessing Technology” by Wulf Harder and Atis Straujums dated Mar. 11,2008, which each are hereby incorporated by reference for all purposesas if fully set forth herein. FIG. 2 illustrates a white-box AESimplementation with encodings on the input of the rounds, i.e., on theinput of the S-boxes. As shown, each of the 16 input bytes are encodedby f_(i) and each of the output bytes are encoded by g_(i).

In order to describe embodiments of the invention, a basic descriptionof a table-based white-box AES implementation will be described. For amore detailed description of a method for implementing a table-basedwhite-box AES see Chow 1. Chow 1 illustrates a specific implementationthat breaks up certain functions using tables of specified sizes. It iswell understood that various other divisions of the tables may be maderesulting in different functions for the look-up tables and differentsizes. Further, while the embodiments of the invention described belowuse a table-based white-box implementation of AES, other ciphers andcryptographic functions may be implemented according to the embodimentsdescribed. Also, other types of white-box implementations may be usedinstead of the table-base implementation, for example, a finite-stateimplementation.

The description of the table-based white-box AES is split into twosteps. In the first step, a round of AES is described as a network oflookup tables. In the second step, the tables are obfuscated by encodingtheir input and output.

Step 1: Implementing AES as a Network of Lookup Tables.

AES operates on data blocks of 16 bytes. These are typically describedas a 4×4 byte matrix, called the state including bytes x_(1,1), x_(1,2),x_(1,3), . . . x_(4,4). A round of AES as described above with respectto FIG. 1 include the following operations: AddRoundKey 110, SubBytes120, ShiftRows 130, and MixColumns 140. The first two operations,AddRoundKey and SubBytes can be merged into a single T-box operation.That is, we can define a byte-to-byte function T_(i,j) for input bytex_(i,j) as T_(i,j)(x_(i,j))=S(x_(i,j)⊕k_(i,j)) where k_(i,j) is a singlebyte of a 16 byte round key based upon the AES key. Let y_(i,j) be theoutput of T_(i,j). The ShiftRows operations is just an index-renumberingof the output bytes y_(i,j). For ease of presentation, this operation isomitted in this description, but may be incorporated into the look-uptable implementing T_(i,j) or implemented as a separate manipulation ofthe state matrix. In the MixColumns step, an output byte z_(i,j) of theround is computed from the 4 output bytes y_(1,j), y_(2,j), y_(3,j), andy_(4,j) via the algebraic expressionz_(l,j)=MC_(l,1)·y_(1,j)⊕MC_(l,2)·y_(2,j)⊕MC_(l,3)·y_(3,j)⊕MC_(l,4)·y_(4,j)in GF(2⁸) for some constants MC_(l,r).

Now define a lookup table for each byte-to-byte functionQ_(i,j,l)(x_(i,j))=MC_(l,i)·T_(i,j)(x_(i,j)) with i, j, l=1, 2, . . . ,16. Then any output byte z_(l,j) may be computed by XORing the resultsof these lookup tables, i.e.,z_(l,j)=Q_(1,j,l)(x_(1,j))⊕Q_(2,j,l)(x_(2,j))⊕Q_(3,j,l)(x_(3,j))⊕Q_(4,j,l)(x_(4,j)).Note that the index i, j, l of Q-box can be interpreted as “thecontribution of input byte i, j of a round to output byte l, j of theround”. The XOR may be implemented to operate on each of two nibbles(i.e., 4-bit values) as a lookup table to reduce the size of the XORtables. Accordingly, the Q-box may be implemented to produce outputnibbles so that the size of the tables is reduced. Therefore, thecomputation of each output byte z_(l,j) of an AES-round has beendescribed as a network of lookup tables. The network of lookup tables tocompute a single output nibble of byte z_(2,3) is shown in FIG. 3.

FIG. 3 illustrates the computation of one output nibble by means of anetwork of look-up tables. The superscript index (1) in the Q-boxesindicates that the tables only provide the first nibble of the output ofthe Q-box. A set of input bytes x_(1,3), x_(2,3), x_(3,3), and x_(4,3)in the input state 310 are input into the Q-boxes 320, 322, 324, 326.The outputs of lookup tables 320 and 322 are fed into the XOR 330, andthe outputs of lookup tables 324 and 326 are fed into the XOR 332. Theoutputs of XORs 330 and 332 are fed into XOR 334. The output of XOR 334is the first nibble of the output z_(2,3) of output state 340. Thesecond nibble of the output z_(2,3) of output state 340 may becalculated in the same way using additional Q-boxes along with a similarXOR network. Further, additional sets of tables may be implemented tocompletely convert the input state 310 into the output state 340 byreceiving a column of bytes from the input state and converting theminto the output of the corresponding column of the output state.

Step 2: Obfuscating the Tables and the Intermediate Values

In the implementation depicted in FIG. 3, the key may easily beextracted from the Q-boxes. Just applying the inverse MixColumnsmultiplication and the inverse S-box to the output reveals the plainAddRoundKey operation. To prevent this, the input and outputs of alllookup tables are encoded with arbitrary bijective functions. This isdescribed in Chow 1. This means that a lookup table is merged with anencoding function that encodes the output and with a decoding functionthat decodes the input. The encodings are chosen such that the outputencoding of one table matches the input encoding assumed in the nexttables. A portion of the implementation of FIG. 3 is depicted in FIG. 4for the first round. In this example, the input to the round is notencoded in order to be compliant with AES, but the output of the roundis encoded. The output encoding is handled in the next round. That is,unlike the first round, the second round (and the later rounds) assumesthat the input is encoded. Alternatively, the first round may receive anencoded input. This input encoding must then be applied elsewhere in thesoftware program containing the white-box implementation. Similarly, thelast round may or may not include an output encoding depending onwhether the output is to be AES compliant. Note that in the white-boximplementation obtained, both the lookup tables and the intermediatevalues are obfuscated.

The description of the table lookup based white-box implementationdescribed above was for the encryption operation of AES. It is notedthat the above description is easily adapted for the decryptionoperation by using the inverse of the SubBytes, ShiftRows, andMixColumns operations (invSubytes, invShiftRows, and invMixColumns).Accordingly, it is assumed that the description above can be used foreither the encryption or decryption operation of AES as needed in theembodiments below.

The related application describes the following embodiments. Let s bethe parameter of a function (e.g., a lookup table) within the white-boximplementation. That is, s is not used for the specification of thefunction, but as parameter of the function. In one embodiment, theinternal encodings of intermediate values in the white-boximplementation may be chosen in dependence of s. In another embodiment adependence on s may be introduced in a computed value in the white-boximplementation, which dependence is annihilated further on in thecomputation so that the correct computed result may be obtained.

The method of introducing the dependence on an arbitrary string s maywork by parameterizing a white-box implementation with s. In atable-based white-box implementation, each lookup table has as inputeither the output (or part of the output) of another lookup table and/orthe input (or part of the input) of the implementation (e.g., theplaintext to be encrypted). Accordingly, lookup tables or functions areintroduced that have string s as their input.

As an example, a single string s including 4 bits may be bound to thewhite-box implementation. The string s may be called a binding stringvalue. This embodiment may easily be extended to larger bit strings byapplying the described method k times resulting in the binding ofstrings of 4k bits.

FIG. 5 illustrates a first embodiment of binding a white-boximplementation. FIG. 5 is similar to FIG. 3, but includes an extensionto include binding the string s to the white-box implementation. Adependence on an arbitrary 4-bit string s may be implemented as follows.Let h₀, . . . , h₁₅ be 2⁴ bijective encoding functions, and let T be a8-to-4-bit lookup table 550 defined by T(v,σ)=h_(σ)(v) where v is anoutput nibble from the Q-box 520 and the nibble σ is an identifyingstring value. As can be seen in FIG. 5, the lookup table T 550 receivesthe input v from Q-box 520 Q⁽¹⁾ _(1,3,2′) and the output of T 550 isinput into the succeeding XOR-table 530. The XOR-table 530 maycompensate for effect of T 550 on v when σ=s. This may be accomplishedby decoding the input to the XOR 530 using the function h_(s) ⁻¹. Whenσ=s this results in the correct value of v being input to the XOR 530.Otherwise, the value input to the XOR 530 is incorrect and results in anincorrect output of the white-box implementation. It is noted that theearlier the lookup table T is implemented in the white-boximplementation, the greater change it will have to the output when σ≠s.

FIG. 6 illustrates the application of obfuscation to the white-boximplementation of FIG. 5. Each of the Q-boxes 620, 622, 624, 626correspond to the Q-boxes in FIG. 5 but include input decodings g_(i)and output encodings f_(i) as shown. Further, the lookup table 650corresponds to the lookup table 550 in FIG. 5 but includes inputdecoding f₁ ⁻¹ and output encoding f_(g). Finally, the lookup XOR-tables630, 632, 634 correspond to the XOR-tables 530, 532, 534 in FIG. 5 butinclude input decodings f_(i) ⁻¹ and output encodings f_(i).

Now a description of extending the embodiment of FIG. 5 to includemultiple binding strings will now be provided. Let W⊆2⁴ be set of 4-bitstrings, and let s∈W. Then the embodiment of FIG. 5 may be extended sothat if s∉W the white-box implementation produces an incorrect result.Define h₀, . . . , h₁₅ such that if i∈W then h_(i)(v)=h_(s)(v) for allnibbles v, and if i∉W then there exists at least one nibble v withh_(i)(v)≠h_(s)(v). This will result in the correct functioning of thewhite-box implementation dependent on the availability of any stringsfrom a set IF.

This functionality may be implemented by modifying the white-boximplementation described in FIG. 5 as follows. The table T 550 may bemodified so that for any identifying string value σ that is in W, h_(σ)is defined such that h_(σ)(v)=h_(s)(v) for all nibbles v, On the otherhand, if the identifying string value σ is not in W, there exists atleast one nibble v with h_(i)(v)≠h_(s)(v). This will lead to thewhite-box implementation functioning incorrectly when the value of σ isnot in W.

FIG. 7 illustrates a second embodiment of binding a white-boximplementation as described in the related application. In thisembodiment, an output of a first function in the white-boximplementation may be perturbed based upon σ or s. Then in an output ofa second function, the perturbation may be compensated using s or σrespectively. Let v₁, v₂, v₃, v₄ be the 4 nibbles computed by the 4Q-tables 720, 722, 724, 726. To the value v₁ computed after lookup tableQ⁽¹⁾ _(1,3,2) 720 a value h(σ,v₁) is added via a lookup table T₁ 750,where h is an arbitrary function with an 8-bit input and a 4-bit output.Hence, T₁(σ,v₁)=v₁⊕h(σ,v₁). The idea is now how to compensate for theaddition of h(σ,v₁) when σ=s. This may be accomplished as follows. Afteradding T₁ 750, the network computes the value v₁⊕v₂⊕v₃⊕v₄⊕h(σ,v₁) whilev₁⊕v₂⊕v₃⊕v₄ should be computed. To compensate for this, a lookup tableT₂ 752 and a XOR lookup table 736 may be added, where T₂ computes thevalue h(s,v₁) and the XOR table adds h(s,v₁) to v₁⊕v₂⊕v₃⊕v₄⊕h(σ,v₁).This gives an implementation in which it is guaranteed that the propervalue is calculated if and only if the implementation receives theparameter σ=s. Again, the lookup tables in FIG. 7 may be obfuscated asdescribed above to obtain a final white-box implementation.

Now a description of extending the embodiment of FIG. 7 to includemultiple binding strings will now be provided. Suppose that thewhite-box implementation is to work correctly for any string σ∈W with 1Vcontaining multiple 4-bit strings. In the above embodiment h is definedas an arbitrary function from 8 to 4 bits. Now define h as an arbitraryfunction with the property that h(v,σ)=h(v,s) for all σ∈W andh(v,σ)≠h(v,s) otherwise. This results in the property that the white-boximplementation works properly for the entire message space if and onlyif σ∈W.

This functionality may be implemented by modifying the white-boximplementation described in FIG. 7 as follows. The table T₁ 750 may bemodified so that for any identifying string value σ that is in W, h_(σ)is defined such that h(v,σ)=h(v,s) for all nibbles v, On the other hand,if the identifying string value σ is not in W, then h(v,σ)≠h(v,s). Thiswill lead to the white-box implementation functioning incorrectly whenthe value of σ is not in V.

Michiels describes a method of using a binding string s to specify aportion of a look up table or software codes used in the white-boximplementation. Accordingly, the white-box implementation only workscorrectly when a string σ equal to the binding string s is present. Thenthe lookup table or software code will function properly. Otherwiseimproper output values may be obtained when the string σ does not equalthe binding string s.

Now to extend the embodiment of Michiels to accommodate a plurality ofbinding strings s, let W⊆2⁴ be set of strings, and let s∈W. Then thecomputation of the white-box implementation is disturbed if s∉W. One mayuse techniques similar to those described above to achieve a similarresult using the embodiment of Michiels.

A method according to the embodiments of the invention may beimplemented on a computer as a computer implemented method. Executablecode for a method according to the invention may be stored on a computerprogram medium. Examples of computer program media include memorydevices, optical storage devices, integrated circuits, servers, onlinesoftware, etc. Accordingly, a white-box system may include a computerimplementing a white-box computer program. Such system, may also includeother hardware elements including storage, network interface fortransmission of data with external systems as well as among elements ofthe white-box system.

In an embodiment of the invention, the computer program may includecomputer program code adapted to perform all the steps of a methodaccording to the invention when the computer program is run on acomputer. Preferably, the computer program is embodied on anon-transitory computer readable medium.

Further, because white-box cryptography is often very complicated and/orobfuscated it is tedious for a human to write. It is therefore ofadvantage to have a method to create the cryptographic system accordingto the embodiments of the invention in an automated manner.

A method of creating the cryptographic system according to the inventionmay be implemented on a computer as a computer implemented method, or indedicated hardware, or in a combination of both. Executable code for amethod according to the invention may be stored on a computer programmedium. In such a method, the computer program may include computerprogram code adapted to perform all the steps of the method when thecomputer program is run on a computer. The computer program is embodiedon a non-transitory computer readable medium.

FIG. 8 is a flow chart illustrating a method of binding a white-boximplementation to a binding string. First, the method begins at 805.Then, a white-box implementation of the cryptographic operation isproduced or received 810. This may be produced as described above usingvarious methods and implementations. Next, information identifying a setof strings such that s∈W may be received 815. The strings s may providebinding to a set of specific software instances or a specific hardwaresystems. Then, the white-box implementation may be modified based uponthe set of strings s 820 so that: when an input σ is received that isone of the set of binding string values, the white-box implementationproduces correct output for the cryptographic operation implemented bythe white-box implementation; and when an input u is received that isnot one of the set of binding string values, the white-boximplementation produces an incorrect output for the cryptographicoperation implemented by the white-box implementation. The variousembodiments discussed above provide different ways in which thiscapability may be implemented in the white-box implementation. Themethod may then end 825.

The cryptographic system described herein may be implemented on a userdevice such as a mobile phone, table, computer, set top box, smart TV,etc. A content provider, such as a television network, video streamservice, financial institution, music streaming service, etc., mayprovide software to the user device for receiving encrypted content fromthe content provider. That software may have the encryption key embeddedtherein as described above, and may also include binding strings asdescribed above. Then the content provider may send encrypted content tothe user device, which may then decrypt using the supplied software anduse the content.

Any combination of specific software running on a processor to implementthe embodiments of the invention, constitute a specific dedicatedmachine.

As used herein, the term “non-transitory machine-readable storagemedium” will be understood to exclude a transitory propagation signalbut to include all forms of volatile and non-volatile memory. Further,as used herein, the term “processor” will be understood to encompass avariety of devices such as microprocessors, field-programmable gatearrays (FPGAs), application-specific integrated circuits (ASICs), andother similar processing devices. When software is implemented on theprocessor, the combination becomes a single specific machine.

It should be appreciated by those skilled in the art that any blockdiagrams herein represent conceptual views of illustrative circuitryembodying the principles of the invention.

Although the various exemplary embodiments have been described in detailwith particular reference to certain exemplary aspects thereof, itshould be understood that the invention is capable of other embodimentsand its details are capable of modifications in various obviousrespects. As is readily apparent to those skilled in the art, variationsand modifications can be effected while remaining within the spirit andscope of the invention. Accordingly, the foregoing disclosure,description, and figures are for illustrative purposes only and do notin any way limit the invention, which is defined only by the claims.

What is claimed is:
 1. A non-transitory machine-readable storage mediumencoded with instructions for execution by a white-box implementation ofa cryptographic function having a plurality of layers with a statematrix configured to indicate the state of the cryptographic operationbetween layers, wherein the cryptographic function maps an input messageinto an output message comprising instructions for implementing: a firstsubstitution box in a first layer configured to receive a first portionof the state matrix and produce a first output; a second substitutionbox in a first layer configured to receive a second portion of the statematrix and produce a second output; a lookup table configured to receivethe first output and an identifying string value to produce an encodedfirst output, wherein the encoding of the first output is selected basedupon the identifying string; an XOR configured to decode the encodedfirst output based on a set of binding string values and to combine thedecoded first output and the second output, wherein the decoded firstoutput is equal to the first output when the identifying string value isequal to one of a set of binding string values resulting in the outputmessage having a correct value, and wherein the decoded first output isnot equal to the first output when the identifying string value is notequal to one of a set of binding string values resulting in the outputmessage having an incorrect value.
 2. The non-transitorymachine-readable storage medium encoded with instructions for executionby a white-box implementation of a cryptographic function of claim 1,wherein the identifying string value is based upon an identification ofthe white-box implementation of a cryptographic function.
 3. Thenon-transitory machine-readable storage medium encoded with instructionsfor execution by a white-box implementation of a cryptographic functionof claim 1, wherein the identifying string value is based upon a hash ofa portion of code in the white-box implementation of a cryptographicfunction.
 4. The non-transitory machine-readable storage medium encodedwith instructions for execution by a white-box implementation of acryptographic function of claim 1, wherein the identifying string valueis based upon an identification of a cryptographic system implementingthe white-box implementation of the cryptographic function.
 5. Thenon-transitory machine-readable storage medium encoded with instructionsfor execution by a white-box implementation of a cryptographic functionof claim 1, wherein the identifying string value is based upon a userpassword.
 6. The non-transitory machine-readable storage medium encodedwith instructions for execution by a white-box implementation of acryptographic function of claim 1, wherein the white-box implementationof the cryptographic function includes a network of finite statemachines.
 7. The non-transitory machine-readable storage medium encodedwith instructions for execution by a white-box implementation of acryptographic function of claim 1, wherein the white-box implementationof the cryptographic function includes a network of lookup tables. 8.The non-transitory machine-readable storage medium encoded withinstructions for execution by a white-box implementation of acryptographic function of claim 1, wherein the cryptographic function isone of advanced encryption system (AES) or data encryption standard(DES).
 9. A non-transitory machine-readable storage medium encoded withinstructions for execution by a white-box implementation of acryptographic function having a plurality of layers with a state matrixconfigured to indicate the state of the cryptographic operation betweenlayers, wherein the cryptographic function maps an input message into anoutput message comprising instructions for implementing: a firstsubstitution box in a first layer configured to receive a first portionof the state matrix and produce a first output v₁; a second substitutionbox in a first layer configured to receive a second portion of the statematrix and produce a second output v₂; a first lookup table configuredto receive the first output v₁ and an identifying string value σ tocalculate v₁⊕h(σ,v₁), wherein h is an arbitrary function; a secondlookup table configured to receive the first output v₁ to calculateh(s,v₁) where s is a set of binding string values; a first XORconfigured to calculate v₁⊕h(σ,v₁)⊕v₂; and a second XOR configured tocombine v₁⊕h(σ,v₁)⊕v₂ with h(s,v₁), wherein the output of the second XORis based on v₁⊕v₂ when the identifying string value σ is equal to one ofa set of binding string values s resulting in the output message havinga correct value, and wherein the output of the second XOR is not basedon v₁⊕v₂ when the identifying string value σ is not equal to one of aset of binding string values s resulting in the output message having anincorrect value.
 10. The non-transitory machine-readable storage mediumencoded with instructions for execution by a white-box implementation ofa cryptographic function of claim 9, wherein the identifying stringvalue is based upon an identification of the white-box implementation ofa cryptographic function.
 11. The non-transitory machine-readablestorage medium encoded with instructions for execution by a white-boximplementation of a cryptographic function of claim 9, wherein theidentifying string value is based upon a hash of a portion of code inthe white-box implementation of a cryptographic function.
 12. Thenon-transitory machine-readable storage medium encoded with instructionsfor execution by a white-box implementation of a cryptographic functionof claim 9, wherein the identifying string value is based upon anidentification of a cryptographic system implementing the white-boximplementation of the cryptographic function.
 13. The non-transitorymachine-readable storage medium encoded with instructions for executionby a white-box implementation of a cryptographic function of claim 9,wherein the identifying string value is based upon a user password. 14.The non-transitory machine-readable storage medium encoded withinstructions for execution by a white-box implementation of acryptographic function of claim 9, wherein the white-box implementationof the cryptographic function includes a network of finite statemachines.
 15. The non-transitory machine-readable storage medium encodedwith instructions for execution by a white-box implementation of acryptographic function of claim 9, wherein the white-box implementationof the cryptographic function includes a network of lookup tables. 16.The non-transitory machine-readable storage medium encoded withinstructions for execution by a white-box implementation of acryptographic function of claim 9, wherein the cryptographic function isone of advanced encryption system (AES) or data encryption standard(DES).
 17. A method of mapping an input message into an output messageusing a white-box implementation of a cryptographic function having aplurality of layers with a state matrix configured to indicate the stateof the cryptographic operation between layers, comprising: receiving afirst portion of the state matrix by a first substitution box in a firstlayer and producing a first output; receiving a second portion of thestate matrix by a second substitution box in a first layer and producinga second output; producing an encoded first output by a first a lookuptable configured to receive the first output and an identifying stringvalue, wherein the encoding of the first output is selected based uponthe identifying string; decoding the encoded first output, by an XOR,based on a set of binding string values and combining the decoded firstoutput and the second output, wherein the decoded first output is equalto the first output when the identifying string value is equal to one ofa set of binding string values resulting in the output message having acorrect value, and wherein the decoded first output is not equal to thefirst output when the identifying string value is not equal to one of aset of binding string values resulting in the output message having anincorrect value.
 18. The method of claim 17, wherein the identifyingstring value is based upon an identification of the white-boximplementation of a cryptographic function.
 19. The method of claim 17,wherein the identifying string value is based upon a hash of a portionof code in the white-box implementation of a cryptographic function. 20.The method of claim 17, wherein the identifying string value is basedupon an identification of a cryptographic system implementing thewhite-box implementation of the cryptographic function.
 21. The methodof claim 17, wherein the identifying string value is based upon a userpassword.
 22. The method of claim 17, wherein the white-boximplementation of the cryptographic function includes a network offinite state machines.
 23. The method of claim 17, wherein the white-boximplementation of the cryptographic function includes a network oflookup tables.
 24. The method of claim 17, wherein the cryptographicfunction is one of advanced encryption system (AES) or data encryptionstandard (DES).
 25. A method of mapping an input message into an outputmessage using a white-box implementation of a cryptographic functionhaving a plurality of layers with a state matrix configured to indicatethe state of the cryptographic operation between layers, comprising:receiving a first portion of the state matrix by a first substitutionbox in a first layer and producing a first output v₁; receiving a secondportion of the state matrix by a second substitution box in a firstlayer and producing a second output v₂; calculating v₁⊕h(σ,v₁) by afirst lookup table configured to receive the first output v₁ and anidentifying string value σ to, wherein h is an arbitrary function;receiving the first output v₁ by a second lookup table configured tocalculate h(s,v₁) where s is a set of binding string values; calculatingv₁⊕h(σ,v₁)⊕v₂ by a first XOR; and combining v₁⊕h(σ,v₁)⊕v₂ with h(s,v₁)by a second XOR, wherein the output of the second XOR is based on v₁⊕v₂when the identifying string value σ is equal to one of a set of bindingstring values s resulting in the output message having a correct value,and wherein the output of the second XOR is not based on v₁⊕v₂ when theidentifying string value σ is not equal to one of a set of bindingstring values s resulting in the output message having an incorrectvalue.
 26. The method of claim 25, wherein the identifying string valueis based upon an identification of the white-box implementation of acryptographic function.
 27. The method of claim 25, wherein theidentifying string value is based upon a hash of a portion of code inthe white-box implementation of a cryptographic function.
 28. The methodof claim 25, wherein the identifying string value is based upon anidentification of a cryptographic system implementing the white-boximplementation of the cryptographic function.
 29. The method of claim25, wherein the identifying string value is based upon a user password.30. The method of claim 25, wherein the white-box implementation of thecryptographic function includes a network of finite state machines. 31.The method of claim 25, wherein the white-box implementation of thecryptographic function includes a network of lookup tables.
 32. Themethod of claim 25, wherein the cryptographic function is one ofadvanced encryption system (AES) or data encryption standard (DES).